化工应用数学9664.pptx

《化工应用数学9664.pptx》由会员分享，可在线阅读，更多相关《化工应用数学9664.pptx（36页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

1、化工應用數學化工應用數學授課教師：授課教師：郭修伯郭修伯 助理教授助理教授Lecture 3應用數學方程式表達物理現象建立數學模式建立數學模式lThe conservation lawsmaterial balanceheat balanceenery balancelRate equationsthe relationship between flow rate and driving force in the field of fluid flowheat transferdiffusion of matter建立數學模式建立數學模式lThe conservation lawsmateri

2、al balanceheat balanceenery balancel(rate of)input-(rate of)output=(rate of)accumulation範例說明範例說明A single-stage mixer settler is to be used for the continuous extractionof benzoic acid from toluene,using water as the extracting solvent.The two streams are fed into a tank A where they are stirred vigo

3、rously,and the mixture is then pumped into tank B where it is allowed to settleinto two layers.The upper toluene layer and the lower water layer areremoved separately,and the problem is to find what proportion of thebenzoic acid has passed into the solvent phase.watertoluene+benzoic acidtoluene+benz

4、oic acidwater+benzoic acid簡化（理想化）簡化（理想化）S m3/s tolueney kg/m3 benzoic acidR m3/s toluenex kg/m3 benzoic acidS m3/s waterR m3/s toluenec kg/m3 benzoic acidRate equation for the extraction efficiency:y=mxMaterial Balance:Input of benzoic acid=output of benzoic acidRc=Rx+SySame method can be applied to

5、 multi-stages.隨時間變化隨時間變化Funtion of time非穩定狀態非穩定狀態(unsteady state)In unsteady state problems,time enters as a variable and someproperties of the system become functions of time.Similar to the previous example,but now assuming that the mixer isso efficient that the compositions of the two liquid strea

6、ms are inequilibrium at all times.A stream leaving the stage is of the samecomposition as that phase in the stage.The state of the system at a general time t,wher x and y are now functions of time.S m3/s tolueney kg/m3 benzoic acidR m3/s toluenex kg/m3 benzoic acidS m3/s waterR m3/s toluenec kg/m3 b

7、enzoic acidV1,xV2,yMaterial balance on benzoic acidS m3/s tolueney kg/m3 benzoic acidR m3/s toluenex kg/m3 benzoic acidS m3/s waterR m3/s toluenec kg/m3 benzoic acidV1,xV2,yInput-output=accumulation單位時間的變化t=0,x=0Mathematical ModelslSalt accumulation in a stirred tankt=0Tank contains 2 m3 of waterQ:D

8、etermine the salt concentration in the tankwhen the tank contains 4 m3 of brineBrineconcentration 20 kg/m3feed rate 0.02 m3/sFlow0.01 m3/s建立數學模式建立數學模式lV and x are function of time tlDuring t:balance of brinebalance of saltBrineconcentration 20 kg/m3feed rate 0.02 m3/sBrine0.01 m3/sV m3x kg/m3解數學方程式解

9、數學方程式lSolvelx=20-20(1+0.005 t)-2lV=2+0.01 tMathematical ModelslMixingPure water3 l/minMixture2 l/minMixture3 l/minMixture4 l/minMixture1 l/minTank 1Tank 2t=0Tank 1 contains 150 g of chlorine dissolved in 20 l waterTank 2 contains 50 g of chlorine dissolved in 10 l waterQ:Determine the amount of chlo

10、rine in each tank at any time t 0建立數學模式建立數學模式lLet xi(t)represents the number of grams of chlorine in tank i at time t.lTank 1:x1(t)=(rate in)-(rate out)lTank 2:x2(t)=(rate in)-(rate out)lMathematical model:x1(t)=3*0+3*x2/10-2*x1/20-4*x1/20 Pure water3 l/minMixture2 l/minMixture3 l/minMixture4 l/minM

11、ixture1 l/minTank 1Tank 2x2(t)=4*x1/20-3*x2/10-1*x2/10 解數學方程式解數學方程式lHow to solve?lUsing MatriceslX=AX;X(0)=X0 where x1(t)=120e-t/10+30e-3t/5x2(t)=80e-t/10-30e-3t/5Mathematical ModelslMass-Spring SystemSuppose that the upper weight is pulled down one unit and the lower weight is raised one unit,then

12、both weights are released from rest simultaneously at time t=0.Q:Determine the positions of the weights relative totheir equilibruim positions at any time t 0k1=6k3=3k2=2m1=1m2=1y2y1建立數學模式建立數學模式lEquation of motionlweight 1:lweight 2:lMathematical model:m1 y1”(t)=-k1 y1+k2(y2-y1)k1=6k3=3k2=2m1=1m2=1y

13、2y1m2 y2”(t)=-k2(y2-y1)-k3 y2 解數學方程式解數學方程式lHow to solve?y1(t)=-1/5 cos(2t)+6/5 cos(3t)y2(t)=-2/5 cos(2t)-3/5 cos(3t)隨位置變化隨位置變化Funciotn of positionMathematical ModelslRadial heat transfer through a cylindrical conductorTemperature at a is ToTemperature at b is T1Q:Determine the temperature distributi

14、onas a function of r at steady staterr+drab建立數學模式建立數學模式lConsidering the element with thickness rlAssuming the heat flow rate per unit area=QlRadial heat fluxlA homogeneous second order O.D.E.where k is the thermal conductivity解數學方程式解數學方程式lSolve流場流場(Flow systems)-EulerianlThe analysis of a flow syste

15、m may proceed from either of two different points of view:Eulerian methodlthe analyst takes a position fixed in space and a small volume element likewise fixed in spacelthe laws of conservation of mass,energy,etc.,are applied to this stationary systemlIn a steady-state condition:the object of the an

16、alysis is to determine the properties of the fluid as a function of position.流場流場(Flow systems)-Lagrangianthe analyst takes a position astride a small volume element which moves with the fluid.In a steady state condition:lthe objective of the analysis is to determine the properties of the fluid comp

17、rising the moving volume element as a function of time which has elapsed since the volume element first entered the system.lThe properties of the fluid are determined solely by the elapsed time(i.e.the difference between the absolute time at which the element is examined and the absolute time at whi

18、ch the element entered the system).In a steady state condition:lboth the elapsed time and the absolute time affect the properties of the fluid comprising the element.Eulerian 範例範例A fluid is flowing at a steady state.Let x denote the distance from theentrance to an arbitrary position measured along t

19、he centre line in thedirection of flow.Let Vx denote the velocity of the fluid in the x direction,A denote the area normal to the x direction,and denote thefluid density at point x.Apply the law of conservation of mass to an infinitesimal element of volume fixed in space and of length dx.xdx,A,Vx+d,

20、A+dA,Vx+dVxxdx,A,Vx+d,A+dA,Vx+dVxIf Vx and are essentially constant across the area A,The rate of input of mass is:The rate of mass output is:Rate of input-rate of output=rate of accumulation0Equation of continuityLagrangian 範例範例Consider a similar system.An infinitesimal volume element whichmoves wi

21、th the fluid through the flow system.Let denote the elapsed time:=t-t0where t is the absolute time at which the element is observed andt0 is the absolute time at which the element entered the system.At elapsed time,the volume of the element is Aa,the density is,and the velocity of the element relati

22、ve to the stationary wall is Vx.Apply the law of conservation of mass to the volume element.xa,A,Vxxa,A,Vxt integralxThe elapsed time:The difference between the relative velocity of the forward face and the relativevelocity of the trailing face is the change rate of the length of the element:Mass ba

23、lance of the element at steady-state 和Eulerian結果一樣獨立參數獨立參數(independent variable)lThese are quantities describing the system which can be varied by choice during a paticular experiment independently of one another.lExamples:timecoordinates非獨立參數非獨立參數(dependent variable)lThese are properties of the sys

24、tem which change when the independent variables are altered in value.There is no direct control over a dependent variable during an experiment.lThe relationship between independent and depend variables is one cause and effect;the independent variable measures the cause and the depend variable measur

25、es the effect of a particular action.lExamples:temperatureconcentrationefficiency變數變數(Parameter)lIt consists mainly of the charateristics properties of the apparatus and the physical properties of the materials.lIt contains all properties which remain constant during an individual experiment.However

26、,a different constant value can be taken by a property during different experiments.lExamplesoverall dimensions of the apparatusflow rateheat transfer coefficientthermal conductivitydensityinitial or boundary values of the depent variables各符號之間的關係各符號之間的關係lA dependent variable is usually differentiat

27、ed with respect to an independent variable,and occasionally with respect to a parameter.lWhen a single independent variable is involved in the problem,it gives rise to ordinary differential equations.lWhen more than one independent variable is needed to describe a system,the usual result is a partia

28、l differential equation.邊界條件邊界條件(Boundary conditions)lThere is usually a restriction on the range of values which the independent variable can take and this range describes the scope of the problem.lSpecial conditions are placed on the dependent variable at these end points of the range of the indep

29、endent varible.These are natually called“boundary conditions”.常見的邊界條件常見的邊界條件l熱傳熱傳(heat transfer)Boundary at a fixed temperature,T=T0.Constant hear flow rate through the boundary,dT/dx=A.Boundary thermally insulated,dT/dx=0.Boundary cools to the surroundings through a film resistance described by a h

30、eat transfer coefficient,k dT/dx=h(T-T0).lk is the thermal conductivity;h is the heat transfer coefficient;and T0 is the temperature of the surrendings.邊界值與起始值邊界值與起始值(Boundary value and initial value)lSpecifying conditions on a solution and its derivative at the ends of an interval(boundary value pr

31、oblem)is quite different from specifying the value of a solution and its derivative at a given point(initial value problem).lBoundary value problems usually do not have unique solutions,and it is this lack of uniqueness which makes certain boundary value problems important in solving P.D.E.of physic

32、s and engineering.心得心得l如何建立一個應用數學問題？如何建立一個應用數學問題？由假設，將問題簡化。確定所要探討的目標，找出非獨立參數。例如溫度、濃度等。找出獨立參數，使得非獨立參數可經由獨立參數表示。例如位置、時間等。找出可將獨立參數 及非獨立參數的關係經由數學式表示出的變數。例如氣體流速、熱傳係數等。選定一個特殊點，應用非獨立參數來描述該系統的狀態。增加微量非獨立參數。應用泰勒展開式來表示該微量增加後，該系統的狀態。應用守恆定律或速率方程式來顯示增加的微量。將增加的微量取極限值，建立該模型方程式。將邊界條件確定。谢谢观看/欢迎下载BY FAITH I MEAN A VISION OF GOOD ONE CHERISHES AND THE ENTHUSIASM THAT PUSHES ONE TO SEEK ITS FULFILLMENT REGARDLESS OF OBSTACLES.BY FAITH I BY FAITH